Microlocalization of resonant states and estimates of the residue of the scattering amplitude

Jean-François Bony; Laurent Michel

Displaying similar documents to “Microlocalization of resonant states and estimates of the residue of the scattering amplitude”

Scattering amplitude for the Schrödinger equation with strong magnetic field

Laurent Michel (2005)

Journées Équations aux dérivées partielles

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In this note, we study the scattering amplitude for the Schrödinger equation with constant magnetic field. We consider the case where the strengh of the magnetic field goes to infinity and we discuss the competition between the magnetic and the electrostatic effects.

Weyl type upper bounds on the number of resonances near the real axis for trapped systems

Plamen Stefanov (2001)

Journées équations aux dérivées partielles

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We study semiclassical resonances in a box $Ω (h)$ of height $h^{N}$ , $N ≫ 1$ . We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $𝒯$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^{#} (h)$ with discrete spectrum the number of resonances in $Ω (h)$ is bounded by the number of eigenvalues of $P^{#} (h)$ in an interval a bit larger than the projection of $Ω (h)$ on the real line. As an application,...

Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

Jean-François Bony, Setsuro Fujiié, Thierry Ramond, Maher Zerzeri (2011)

Annales de l’institut Fourier

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We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$ , and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually,...

Resonances for Schrödinger operators with compactly supported potentials

T. J. Christiansen, P. D. Hislop (2008)

Journées Équations aux dérivées partielles

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We describe the generic behavior of the resonance counting function for a Schrödinger operator with a bounded, compactly-supported real or complex valued potential in $d \geq 1$ dimensions. This note contains a sketch of the proof of our main results [, ] that generically the order of growth of the resonance counting function is the maximal value $d$ in the odd dimensional case, and that it is the maximal value $d$ on each nonphysical sheet of the logarithmic Riemann surface in the even dimensional...

Asymptotic behavior of regularized scattering phases for long range perturbations

Jean-Marc Bouclet (2002)

Journées équations aux dérivées partielles

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We define scattering phases for Schrödinger operators on $ℝ^{d}$ as limit of arguments of relative determinants. These phases can be defined for long range perturbations of the laplacian; therefore they can replace the spectral shift function (SSF) of Birman-Krein’s theory which can just be defined for some special short range perturbations (we shall recall this theory for non specialists). We prove the existence of asymptotic expansions for these phases, which generalize results on the SSF. ...

Asymptotic expansion in time of the Schrödinger group on conical manifolds

Xue Ping Wang (2006)

Annales de l’institut Fourier

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For Schrödinger operator $P$ on Riemannian manifolds with conical end, we study the contribution of zero energy resonant states to the singularity of the resolvent of $P$ near zero. Long-time expansion of the Schrödinger group $U (t) = e^{- i t P}$ is obtained under a non-trapping condition at high energies.